A C T A U N I V E R S I T A T I S L O D Z I E N S I S
FO LIA O E C O N O M IC A 175, 2004
A n n a S z y m a ń s k a *
SELECTED STATISTICAL METHODS OF INSURANCE RISK ASSESSMENT
Abstract. Effective m anagem ent o f an insurance com pany calls for diverse types o f quantitative inform ation. D ifferent data sets are needed to fix premiums and different ones for loss handling purposes. The m ass character o f insurance contracts causes that we deal with a huge am ount o f data. H ence, the necessity o f applying statistical m ethods in exam ining regularities governing insurance processes. The goal o f this work is to present statistical m ethods m ost frequently used for insurance risk assessm ent, that is for exam ining distributions o f random variables o f the number and size o f claims in the portfolio.
Key works: insurance risk, the number o f claims, approxim ating m ethods.
1. IN T R O D U C T IO N
T he objects o f insurance statistics research are th e insureds set and the insurance accidents set. T h e statistical unit is th e object o f insu ran ce. F o r exam ple, vehicles are statistical units in m o to r insurance.
A set o f statistical units o f a given type is called the in su ran ce p o rtfo lio . A nalysing a statistical unit from the insurance risk assessm ent p o in t, we are interested prim arily in such characteristics as: n u m b er an d frequency o f occurred losses and value o f claim s.
D u e to the fact th a t statistical estim ates are m ad e on th e basis o f historical d a ta , statistical m eth o d s in the case o f new insurance p ro d u c ts suffer from som e co n stra in ts, w hich are aggravated by a s h o rt h isto ry o f the Polish insu ran ce m ark e t.
S tatistical m eth o d s o f insurance risk assessm ent aim a t d eterm ining d istrib u tio n s o f ra n d o m variables o f the n um b er and size o f claim s and their m ain param eters. T h re e basic g ro ups o f statistical m eth o d s used for insurance risk assessm ent can be distinguished:
descriptive statistics m eth o d s used for estim ating the em pirical d ist ribution function and its m ain p aram eters,
analytical m eth o d s o f estim ating the rand om variable d istrib u tio n function fitted to real d a ta ,
— m ethod of estim ating only m ain characteristics of the ran d o m variable.
2. PR O BA BILITY D IST R IB U T IO N OF T H E N U M BER O F C L A IM S
II the period o f tim e, d u rin g which losses occur is fixed, then the num ber o f claim s causcd by a given entity or the p o rtfo lio o f risks is usually a discrete ra n d o m variable. In the actuarial practice there is m ost frequently used the em pirical distribu tion function, with p ro babilities being estim ated by m eans o f observed frequencies, with which the values o f the rand om variable have been tak en ( B o w e r s et. al. 1986). H ow ever, the past is no t always representative for the future. In such cases th e distrib u tio n of the ra n d o m variable of the n u m b er o f claim s is sought. In the case when the n um ber of losses is, m oreover, a function o f tim e, wc can speak a b o u t a discrete ra n d o m proccss.
In practice we can m eet po rtfolios com posed o f a big n u m b er o f individual risks characterised by small probabilities o f loss occurrcnce. T h en the process o f N ( t ) losses in the tim e period from 0 to í is P oisson distributed on co n d itio n the follow ing assum ption s are fulfilled:
num bers o f losses o ccurrin g in any tw o disjoint tim e intervals arc independent;
no m ore th an one claim can arise from the sam e event;
- probability, th at a loss will occur at a definite time point is equal to zero. D e n o tin g the claim n u m b e r ra n d o m v ariable by N , p ro b a b ility o f exactly n claim s occurrence in a given period o f tim e am o u n ts to:
A"
P ( N = и) = e for и = 0, 1, 2, ... (l)
with A(t) = E[JV].
I roperties of Poisson d istribution indicate th at the num ber o f independent Poisson distrib u ted ra n d o m variables N t , N 2, N m is a ra n d o m variable having P oisson d istrib u tio n with p aram eter A = At + A2 + ... + Am, being the sum o f p aram eters o f respective random variables N ,, JV2, ..., N m. Let G denote the jo in t d istrib u tio n function o f the ran d o m variable N *
Recursion formula for calculating distribution o f the random variable of the number of claims.
O ne o f m eth o d s fo r estim ating probabilities is using th e follow ing recursion form ula:
P( N = n) = ^ • P( N = n — 1) (2) n
with initial value
P( N = 0) = e “ (3)
Normal approximation o f probability distribution o f the claim number random variable.
O n the basis o f the cen tral limit theorem ( D o m a ń s k i , P r u s k a 2000) for large m , the ra n d o m variable o f claim s n u m b er N h aving P oisson d istrib u tio n with p a ra m e te r X = m is the sum o f m in d ep en d en t ra n d o m variables identically P oisson d istributed with p aram eter 1, and it can be appro x im ated by m ean s o f the no rm al distribution:
G(" , = N C i r ) (4) Ans corn he approximation o f probability distribution o f the random variable of the number of claims:
(5)
а д - ^ ( . + о м . г » - ^ + 5 у
Peizer and Pratt probability distribution o f the claim number variable:
W " N ( K + ; M * + ^ ) ] V ' +T(2)) <6> where n + 0.5 Z = — (7)
t í \
i
- 22
+
2
zH z)
,Q4 T(z) = --- --- (8)A p p ro x im atio n fo rm u las have been com pared w ith one a n o th e r in the T ab. 1.
T a b l e 1
Com parison o f the value o f random variable o f the number o f claims estimated by different m ethods
X n
Values o f distribution function G(n) for n < X and 1 — G(n) for n > X o f Poisson distributed random variable
exact values normal approxim ation anscom be approxim ation
Peizer and Pratt approxim ation 10 0 0.00045 0.000783 0.000034 0.000044 2 0.002769 0.005706 0.002672 0.002763 8 0.332820 0.263545 0.332775 0.332833 10 0.583040 0.500000 0.582704 0.583059 12 0.208444 0.263545 0.208786 0.208432 18 0.007187 0.000252 0.007137 0.007187 23 0.000120 0.000020 0.000115 0.000120 100 80 0.022649 0.022750 0.022643 0.022649 90 0.171385 0.158655 0.171405 0.171386 100 0.526562 0.500000 0.526551 0.526563 110 0.147137 0.158655 0.147161 0.147137 120 0.022669 0.022750 0.022665 0.022669 130 0.001707 0.001350 0.001703 0.001707 140 0.000064 0.000032 0.000064 0.000064 145 0.000010 0.000003 0.000010 0.000010 1 000 905 0.001215 0.001332 0.001214 0.001215 937 0.023172 0.023173 0.023172 0.023172 968 0.159596 0.155786 0.159599 0.159596 1 032 0.152095 0.155786 0.152097 0.152095 1 063 0.023155 0.023173 0.023155 0.023155 1 095 0.001446 0.001332 0.001446 0.001446 S o u r c e : D a y к i n et at. 1994.
R esults o f num erical tests m ad e by C. D. D ay k in show ed th a t the m axim al e rro r in A nscom be ap p ro x im atio n is low er th a n 1 0 "4 fo r A/35. A c o n stra in t for Peizer and P ra tt m ethod is A/6. N o rm al ap p ro x im a tio n yields good results fo r A/1000. A n ad v an tag e o f Peizer and P ra tt a p p ro x im a tio n fo rm u la is its co rrectn ess for sm all A values. A n sco m b e ap p ro x im a tio n is m o re convenient th an Peizer and P ra tt ap p ro x im a tio n for num erical reasons.
C hanges in intensity o f losses in the portfolio caused by extern al factors such as w eather, econom ic con ditions, and so on are frequ ently observed in practice.
If changes in intensity o f losses arc o f ra n d o m ch a rac te r, th en the random variable o f the num ber o f claims can have Poisson mixed distribution. Mixed Poisson claim n u m b er variable Q > 0 fulfils co n d itio n E[Q] = 1, which m eans th a t the intensity o f losses over a certain tim e period can be at a definite level. I f Q assum es values bigger th a n 1, th en th e intensity o f losses is higher th a n expected, if it assum es values in the interval from
0 to 1, then the intensity o f losses is low er th an expected.
If ra n d o m v a ria b le Q accep ts value q, th en th e c o n d itio n a l claim nu m b er d.f. P ( N / Q = q) is P oisson d istrib u tio n w ith Xq p aram eter.
T h e problem is to estim ate d istrib u tio n o f the m ixing ran d o m variable Q. T o o few d a ta are usually available to co n stru ct an alytically th e form o f the m ixing ran d o m variable d istrib u tio n function. A t such tim e th e m eth o d o f m o m en ts ( D o m a ń s k i 2 0 0 1) is used and only the m ain characteristics arc estim ated w ith o u t seeking the d istrib u tio n functio n form . If the n um b er o f d a ta is big eno u g h frequency series are form ed and the d istrib u tio n function form is estim ated on th eir basis. T h e m ixing ra n d o m v ariab le m o st frequently has gam m a d istrib u tio n ( B o w e r s et al. 1986).
E qually interesting problem is a search for the d istrib u tio n fu n ctio n o f the n u m b er o f claim s com ing from individual insu ran ce policies. T h is is significant d u e to flu c tu a tio n s in o ccurrence o f losses fro m p a rtic u la r policies in the p o rtfo lio , w hich has a direct im pact on the level o f prem ium s.
E ach n th policy has the claim frequency p aram eter k, described by the form ula:
fc, = k h t (9)
where к is th e average n u m b er o f losses, an d th e dev iatio n coefficient per unit from k. D istrib u tio n o f risk in th e p o rtfo lio is ch aracterised by the d istrib u tio n fu nction H o f ran d o m variable hľ F u n c tio n H is called the risk stru ctu re fu n ctio n in the po rtfo lio .
3. PR O B A B IL IT Y D IS T R IB U T IO N OF T H E A M O U N T O F C L A IM S
T h e a m o u n t o f claim s resultin g from an individual loss is a ra n d o m variable o f co n tin u o u s type ( B o w e r s et al. 1986).
T h ree m eth o d s are used for estim ating the d istrib u tio n o f ra n d o m variables fo r an individual loss ( D a y к i n et al. 1994):
- analytical m e th o d ,
- em pirical d istrib u tio n fu nction co n stru c tio n m eth o d , - basic p aram eters estim atio n m eth o d ,
T hese m eth o d s will be briefly discussed below.
T h e an a ly tica l m e th o d consists in ch o o sin g th e an a ly tica l form o f d istrib u tio n fun ction fitting to the observed d ata. T his m eth o d practically m eans ch oosing such a d istrib u tio n am ong know n d istrib u tio n s, which fulfils a definite criterio n best.
T h e m ost frequently used d istrib u tio n s are: rand om variable has: gam m a, logarithm ic o r P areto d istrib u tio n s ( R o n k a - C h m i c l o w i c c 1997).
We choo se the d istrib u tio n , which m inim ises th e value o f / 2 statistics
( D o m a ń s k i 2001).
T h e em pirical d istrib u tio n function co n stru c tio n m eth o d consists in building a disjoint series from observed d a ta and co n stru c tin g the em pirical d istrib u tio n fu nction on this basis.
T h e ta b u la r m eth o d o f the em pirical d istrib u tio n fu nction co n stru c tio n P(x) is com m only used here:
Р ( х ) = ^7Г (10)
where к is the n u m b er o f claim s with values lower o r equal to x and К is the to tal n u m b er o f losses. T h e disjoint scries table is built on th e basis of d a ta and next the em pirical d istrib u tio n function is constructed (D a y k i n et al. 1994).
It should be rem em bered th a t intervals have to be built in an unbiased way and th a t d u e to specificity o f insurance the length o f class intervals should increase along with the num b er o f claim s (for instance, geom etrically).
The ta b u la r m eth o d is used in the case o f a big d a ta n um b er.
T h e m eth o d o f m ain pa ram eters estim ation consists in cho osin g the distrib u tio n function w ith given p aram eters d eterm ining the d istrib u tio n . T h e p aram eters are estim ated by m eans o f the m axim um likelihood m eth o d or the m ethod o f m om ents. Follow ing estim ation o f the theoretical distribution function the fit o f theoretical an d em pirical d istrib u tio n s is tested using x2
or Я-K o lm o gorov tests ( D o m a ń s k i 2 0 0 1).
If for all know n fam ily d istrib u tio n s the zero hypothesis (H 0: the sam ple for which the em pirical d istrib u tio n has been estim ated com es from the p o p u latio n with the tested theoretical d istrib u tio n ) is u n tru e, th en the insurance p o rtfo lio is heterogeneous. In such case the p o rtfo lio sh ould be divided in to risk gro u p s so th a t a uniform d istrib u tio n o f th e n u m b e r o f claim s P , can be fo u n d in each g roup. T hen the d istrib u tio n function P will be a convex co m b in atio n o f d istrib u tio n functions P t w ith a p p ro p ria te weights. H ow ever, finding such weight can be im possible. Som e au th o rs suggest th a t the m eth o d o f limited expected value fu n ctio n should be used in such case ( D a y k i n el al. 1994).
All th ree m eth o d s arc generally used sim ultaneously to estim ate the claim nu m b er d istrib u tio n in the actuarial practice. D a ta are verified in such a way th a t allow ances can be m ade for inflation.
T h e length o f research periods is frequently d ifferentiated acco rd in g to the loss size. D a ta a b o u t large losses, for exam ple, ca ta stro p h es, for w hich the research period should a m o u n t from 10 to 1 0 0 years, arc m o st frequently unavailable D espite the fact th a t these losses exert a significant im pact on the d istrib u tio n fun ction, it is im possible to accept such long research period in the Polish m ark e t.
R EFEREN CES
B o w e r s N. I., G e r b e r H. U. , H i c k m a n J. C., J o n e s D. A. , N e s b i t t C. J. (1986),
A ctuarial M ath em a tics, T he Society o f Actuaries, Itasca (III).
D a y k i n C. D. , P e n t i a i n e n T. , P e s o n e n M. (1994), P ractical R is k T heory f o r A ctuaries, Chapman & Hall, London.
D o m a ń s k i С. (2001), S ta tistica l M ethods, University o f Ł ódź Press, Łódź.
D o m a ń s k i C., P r u s k a K. (2000), N on-C lassical S ta tistica l M ethods, PW E, Warszawa; R o n k a - C h m i e l o w i e c W. (1997), Insurance R isk - A ssessm ent M ethods, A kadem ia
Ekonom iczna, Wrocław.
A n n a S z y m a ń s k a
W Y BR A N E M E T O D Y S T A T Y S T Y C Z N E OCENY RYZYKA U B E Z P IE C Z E N IO W E G O
Prawidłowe zarządzanie towarzystwem ubezpieczeniowym wym aga różnorodnych informacji wartościowych i ilościow ych. Inne dane są potrzebne d o w yznaczania składek, inne dla potrzeb likwidacji szkód. M asow ość ubezpieczeń sprawia, że w przypadku ubezpieczeń m am y d o czynienia z olbrzymią liczbą danych. Stąd potrzeba zastosow ania m etod statystycznych d o badania prawidłowości rządzących procesami ubezpieczeniowymi. Celem pracy jest przedstawienie najczęściej stosow anych m etod statystycznych, służących d o oceny ryzyka ubezpieczeniow ego, czyli d o badania rozkładów zm iennych losow ych liczby roszczeń i wielkości roszczeń w portfelu.
